(2) Any equation expresses simply the relation of the four first operations to one another, and this also holds of those equations even which contain powers, roots, and logarithms, because the latter 3 forms are as yet so specially conceived that we always may and must treat them as if they were the products, quotients, and 'positive or actual numbers which they represent; or, in other words, because powers, roots, and logarithms have been hitherto so specially conceived, that they have not yet appeared as forms per se. (3) Any such equation, precisely because it only enunciates the relation of the operations to one another, holds perfectly independently of what the several letters in it may represent in particular, i. e. it holds for every value of every letter. (4) But we may imagine equations which will not become correct (real) equations until perfectly determinate expressions have been substituted for some letters which occur in it, as x, y, z, &c.; and which therefore do not become correct (real) equations, until these determinate, frequently still unknown expressions, are considered as being represented by x, y, z, &c. Such equations are then called determining equations, because they are usually employed for the purpose of determining the unknown expressions which must be substituted for x, y, z, &c. in order to obtain the (real) correct, or, as it is usually termed, identical equation, i. e. in order to obtain the only equation which exists, and which enunciates the relation of the laws of operation to one another. The letters x, y, z, &c. which represent these determinate values, are then usually called the "unknown expressions." The process which must be applied, in order to obtain from a determining equation the determinate values of the unknown expression, is termed the solution of the determining equation with respect to this unknown expression. Determining equations are divided into algebraical and transcendental, the algebraical being those in which the unknown expression, conceived as being perfectly general, only occurs in the general combinations which have been hitherto considered. And that part of the mathematical analysis, which treats of determining equations and their solutions, is called algebra, and may also be subdivided into lower and higher (first and second parts of) algebra. SECTION 36. Now follows, without further difficulty, the solution of algebraical (determining) equations of the first degree with one or several unknown expressions, and in the latter case the exhibition of the several methods of elimination. Only from a x = b, we may not conclude that x b a when a = O, because (by sect. 16) we may never divide by zero. On the contrary when a = = 0, the equation ax=b becomes 0.x-b that is 0=b, no longer contains the unknown expression x, and is either correct, or contains a contradiction, which announces the incorrectness of the hypothesis which has led to this equation. (Compare sect. 16. Note.) a We may not, namely, say that x = ∞ when in ax=b, a b becomes zero. But if b and a are positive numbers, then x=will be the greater the smaller a becomes, and x will become infinitely great whenever a is infinitely small, provided that the letter b retains the determinate value which it at first possessed. SECTION 37. If we now proceed to the quadratic (determining) equation ax2 + bx + c = 0, we see at once that the pure quadratic equation x2=q, gives the solution x = = √q and therefore allows of no general solution, until we have introduced /q for a general expression q. But if we introduce the square root q for any general q by defining it as a symbol (as a form) endowed with the property that (q) may be interchanged with q itself, it follows immediately from x=(q) that (x − √q). (x + √q)=0, that is x=+√q and x = - q; that is, there are two unequal and more complex forms + √q (that is 0+ √q) and −√q (that is 0-√q) which equally possess this same property, and there are no more than these. Hence the general square root √g represents in general either of two unequal forms, viz. +√g and √9, unless in particular cases a determinate one of them only be represented. The conclusions from this are extremely important, viz. (A) The general square root q has, when q is positive, two values + a and -a (where a is the value of the absolute root q), one of which is positive and the other negative. (B) When q is not positive and not zero, the general square root q remains a root per se, for which no previous (actual) form can be substituted. Every expression which without being general is not actual (i. e. not positive, negative, or zero) is called imaginary. Hence the general square root is in every particular case either an actual expression, or, if an expression per se, imaginary. (C) For the general square roots (and therefore for the imaginary as well as for the actual) the laws of roots in (sect. 27), viz. the formulæ still hold, viz. (I. 1 and 2) perfectly unconditionally, because in (I.) both sides of the equation are single-meaning, while in (1 and 2) both sides of the equation are exactly double-meaning, so that one side is as complete as the other; while (II.) only holds unconditionally when written thus (II.) √(a2)=±a, because then the same forms stand on the right as are represented by the expression on the left. D If Ja is general (i. e. either actual or imaginary,) or if Ja is actual or a imaginary, we may not put (p±q) Ja for p√a±qJa, nor (a) or a for Ja. Ja, nor from Ja = a, and Jaß, conclude that a = - ß, unless we have previously convinced ourselves that a whenever it occurs in one and the same expression, will also always represent one and the same of its two values. In other words: since this form is double-meaning, do not treat it when it occurs several times as if it were_single-meaning, but always remember that although to outward appearance one and the same form, it may nevertheless represent in each case a different one of its two values*. (E) If q is negative and -p, then √q=√-p=√p.√-1, while p is actual; so that every imaginary root of the form Jp may be always reduced to the simpler imaginary root-1. (F) For calculating with imaginary (square) roots, there are of course no other laws or formulæ but those which were given for general (square) roots, namely those summed up in (C), which however must be applied with the proper precautionary rules in (D). Namely, we shall have 7.√-1-2.√-1=5.√√-1, √-1.√=1=(√−1)2 = − 1, only when we have been able so to arrange the calculations before hand that we are perfectly sure, that √1, wherever it occurs, only represents one and the same of the two forms +√-1 and -√1, which have in common with it its own essential property (viz. that when potentiated by 2 the result is -1). (C) In general therefore we must write, agreeably to the formulæ and SECTION 38. The general solution of the general quadratic equation ax2+bx+c=0, The neglect of this simple rule is, as the history of mathematical analysis teaches us, a fruitful source of contradictions, or so-called "paradoxes of calculation." It gives for the unknown expression x, two and no more than two (unequal) values (i. e. forms which when substituted for x make the equation a x2 + bx + c = 0 correct (identical), viz. 0=0). If we now consider this general solution in the particular case when a, b, c are no longer mere supporters of the operations, but are already considered as actual numbers (i. e. as expressions in cyphers of determinate form) then the two values of x are both actual or both imaginary, according as b2-4ac is positive or negative. But in this last case we may write b √4ac-b3 √-1, X= 2 a ㄓ 2 a and we consequently perceive that in this case both values of x be reduced to the form p+q √−1 where P and q are actual. may SECTION 39. From this time forth then the possible particular forms of calculation are not merely real, but also imaginary, the latter being of the form p+q√-1; and in this form we can also include all actual expressions, because we may conceive q=0. We term the form p+q/-1 therefore a general numerical number. We shall always denote by i one and the same of the forms represented by √-1, so that i will then represent the other. If then p+q.i=r+s.i upon the hypothesis that p, q, r, s are all four actual, we shall have severally p=r and q = s*. Hereupon we find that (1) (a+ß. i)±(y+d. i) = (a±y) + (ß ± d). i ; (3) a+ß.i_ay + ßd, By-ad 2 + 23+82 .i; (4) √a+ß.i=±√1⁄2a + ‡ √ a2 + ß3± √− ‡ a+ § √aa + ßa . i, where in (No. 4) the two exterior roots are considered as single meaning and absolute, and both the (+) or both the (-) signs must be taken when ẞ is positive, while the opposite signs must be prefixed to the same two roots, that is (+) to one and (-) to the other, when ẞ is negative. r q-s *Since i would otherwise (=P or be) actual. We sometimes find in analytical writings the equations X-X', and Y=Y' deduced from X+Y.i=X'+Y'.i while X, X', Y, Y' are still general. This is however always incorrect, and we must not be surprised if such a method of proceeding lead to contradictions. It follows from this that all combinations of such actual or imaginary forms will always lead to the same forms. Even if we suppose the coefficients a, b, c in the general quadratic equation ax2 + bx + c = 0, to be of the form p+q.i, it will immediately follow from the above results (Nos. 1-4) that the two values which would result from this quadratic equation for the unknown expression x, can and must assume the same form, i. e. are general numerical numbers. As long, therefore, as we calculate with the means hitherto exhibited we can arrive at no other new particular forms per se, but all expressions which owe their existence to whole real numbers, i. e. all expressions in cyphers may always be reduced expre to the form p+q -1 or p+q.i, where Р and 9 are actual, so that they may also be equal to zero. SECTION 40. We can now proceed to the higher algebraical equations. We first prove in one of the well known methods (1) That every higher equation of the nth degree with general numerical coefficients (i. e. of the form p+q.i) gives one value for the unknown expression x of the same form P+Q.i. (2) That it gives for the unknown expression x, n such values, neither more nor less, which may however in particular cases be all or some equal to one another. (3) That every whole function of x of the nth degree, viz. x+Ax1+B x2¬2 + С x2¬3 + + Px+Q, ... with general numerical coefficients, (i. e. of the form a+ẞ.i) may be always split into a product of n factors (and this in only one manner) of the form (x − a) (x − b) (x − c) (x − d) (x − p) (x − q), where a, b, c, d ... ... P, q are also general numerical expressions (i. e. of the form a + B. i), namely, those values which when substituted for a make the whole function = 0. (4) If the coefficients A, B, C,... P, Q, are all actual, then those among the n factors which are imaginary may be arranged in pairs, so that each such pair may give the product [x − (p + q . i)] • [x-(p-q.i)], that is x2-2px + (p2 + q3). The above whole function of x of the nth degree has therefore none but actual simple factors, or at any rate actual simple and double factors (i. e. factors of the form x2+ rx + s with actual coefficients). The doctrine of the higher equations may now be continued at pleasure, without encountering any particular difficulties; we may now, namely, easily discover the characteristics by which we can ascertain whether some equal values exist for the unknown expressions, or whether all the values are unequal. |